Two's-complement
Two's complement is a mathematical operation on binary numbers, as well as a binary signed number representation based on this operation. Its wide use in computing makes it the most important example of a radix complement.
The two's complement of an N-bit number is defined as the complement with respect to 2N; This is also equivalent to taking the ones' complement and then adding one, since the sum of a number and its ones' complement is all 1 bits. The two's complement of a number behaves like the negative of the original number in most arithmetic, and positive and negative numbers can coexist in a natural way.
In two's-complement representation, positive numbers are simply represented as themselves, and negative numbers are represented by the two's complement of their absolute value; two tables on the right provide examples for N = 3 and N = 8. In general, negation (reversing the sign) is performed by taking the two's complement. This system is the most common method of representing signed integers on computers. An N-bit two's-complement numeral system can represent every integer in the range −(2N − 1) to +(2N − 1 − 1) while ones' complement can only represent integers in the range −(2N − 1 − 1) to +(2N − 1 − 1). The sum of a number and its two's complement will always equal 0 (since the last digit is truncated), and the sum of a number and its one's complement will always equal -0.
The two's-complement system has the advantage that the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers (as long as the inputs are represented in the same number of bits and any overflow beyond those bits is discarded from the result). This property makes the system both simpler to implement and capable of easily handling higher precision arithmetic. Also, zero has only a single representation, eliminating the subtleties associated with negative zero, which exists in ones'-complement systems.
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